What question did this study set out to answer?

The aim is to address the uniqueness of certain polynomials sharing a specific function involving multiplicities.

March 6, 2026Acta Universitatis Sapientiae Mathematica0 citationsOpen Access

Hayman conjecture and uniqueness of some delay-differential polynomials sharing a small function

Key Points

The aim is to address the uniqueness of certain polynomials sharing a specific function involving multiplicities.
Investigated properties of P(f)L(g) and P(g)L(f) when sharing a small function.
Considered L(h) as specific forms such as derivatives and shifts of functions.
Used mathematical analysis to generalize previous results from Gao and Liu.
Established conditions under which P(f)L(g) and P(g)L(f) are unique.
Found that uniqueness holds when both polynomials share a nonzero small function.
Generalized results from recent literature to broader cases.

Abstract

In this paper, we investigate the uniqueness problem of P (f) L (g) and P (g) L (f) when they share (z) counting multiplicities, where L (h) represents any one of h^ (k) (z), \; h (z+c), \; h (z+c) -h (z) and h^ (k) (z+c) where k is a positive integer, c is a nonzero constant and (z) is a nonzero small function with respect to both f (z) and g (z). The results of the paper generalize the recent results of Gao and Liu (Bull Korean Math Soc 59: 155–166, 2022).

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